Political Analysis

Political Analysis Advance Access published online on May 6, 2008
Political Analysis, doi:10.1093/pan/mpn002

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Two Sides of the Same Coin? Employing Granger Causality Tests in a Time Series Cross-Section Framework

M. V. Hood, III

Department of Political Science, 104 Baldwin Hall, University of Georgia, Athens, GA 30602-1615, USA, email: th{at}uga.edu

Quentin Kidd

Department of Government, Christopher Newport University, 1 University Place, Newport News, VA 23606, USA, email: qkidd{at}cnu.edu

Irwin L. Morris

Department of Government and Politics, University of Maryland at College Park, 3140 Tydings Hall, College Park, MD 20742, USA

e-mail: imorris{at}gvpt.umd.edu (corresponding author)

In this paper, we introduce a recently developed methodology for assessing the assumption of causal homogeneity in a time series cross-section Granger framework. Following a description of the procedure and the analytical contexts for which it is appropriate, we implement this new approach to examine the transformation of the post-World War II party system in the South. Specifically, we analyze the causal relationship between black mobilization and GOP growth in the region. We find that black mobilization Granger caused Republican growth throughout the South, whereas Republican growth Granger caused black mobilization only in the deep South. We discuss the substantive significance of our results and conclude with guidelines for the appropriate use of this procedure and suggestions for future extensions of the method.

	1. Introduction

Top 1. Introduction 2. Assessing Causality--The... 3. A Method for... 4. Substantive Application: The... 5. Discussion of Results 6. Future Directions 7. Conclusion Notes References

 
Introduced nearly 40 years ago, Granger models (see Granger 1969) of various types remain the most popular methodology for evaluating the nature of the causal relationship between two variables.¹ Though originally designed for pairs of lengthy time series, Granger tests are increasingly used to evaluate causal relationships in time series cross-section (TSCS) data. The extension of the original Granger methodology to TSCS data has the potential to improve upon the conventional Granger analysis for all the reasons that TSCS analysis is generally preferable to cross-sectional or traditional time series analysis. In general, TSCS models provide a number of improvements over the separate analyses of time series data by cross-section. First, TSCS data allow for considerably more flexibility modeling the behavior of cross-sectional units than conventional time series analysis (see Greene 2000). Second, the TSCS framework allows for the analytical incorporation of significantly more observations (and more degrees of freedom [df]) than would a comparable analysis of individual time series. Finally, and most significantly for our purposes, TSCS Granger tests are significantly more efficient than conventional Granger tests (Hurlin and Venet 2001). The potential benefits of TSCS Granger tests are, therefore, considerable.

However, just as TSCS Granger tests share the benefits of other types of TSCS analysis, they also share an important potential flaw: the inappropriate assumption of causal homogeneity. As is often the case in TSCS analysis, this assumption is largely ignored. Failure to adequately analyze the empirical foundations for this assumption could easily lead to faulty substantive conclusions—inferring a causal relationship in all cross-sections when it is only present in a subset of cross-sections or rejecting the presence of a causal relationship for an entire group of observations when a subset of the sample actually does manifest the hypothesized causal relationship. As it becomes increasingly clear that (often untested) assumptions about the specification of TSCS models may produce quite fragile results (see Beck 2007; Wilson and Butler 2007), the development of a procedure for evaluating the assumption of causal homogeneity in TSCS models would be a valuable methodological innovation. Following a brief discussion of Granger analysis and its extension to TSCS models, we describe just such a procedure—developed in Hurlin and Venet (2001)—for evaluating the causal homogeneity between cross-sections in a TSCS framework. We then present an application of this new methodology in which we examine the causal relationship between black mobilization and Republican growth within the context of the Southern party system. We conclude by identifying other potential applications of this methodology and provide a set of guidelines for the appropriate use of this procedure in the research process.

	2. Assessing Causality—The Granger Framework

Top 1. Introduction 2. Assessing Causality--The... 3. A Method for... 4. Substantive Application: The... 5. Discussion of Results 6. Future Directions 7. Conclusion Notes References

 
Granger testing is a common method of investigating causal relationships (Granger 1969). By estimating an equation in which y is regressed on k lagged values of y and k lagged values of an additional variable x, we can evaluate the null hypothesis that x does not Granger cause y.² If one or more of the lagged values of x is significant, we are able to reject the null hypothesis that x does not Granger cause y. In more straightforward terms, we conclude that the evidence suggests that x causes y.

The Granger procedure is imperfect. Aside from the fact that no technique can fully ascertain the nature of a causal relationship between two variables, the Granger procedure is a simple bivariate model. It is possible, for example, that the relationship between x and y is spurious (the correlation being caused by the impact of some other variable, z, on x and y), and we cannot evaluate this possibility in the context of a simple bivariate model. This renders the isolated use of Granger tests problematic. Conventional Granger tests are also less than ideal for estimating the general causal relationship between two time series in a variety of independent settings (or cross-sections). Estimating independent Granger models for each of a set of cross-sections and then generalizing from frequently disparate results is an inefficient—though common—procedure to test for the presence of a universal causal relationship between variables. It does not, however, obviate the usefulness of Granger tests as a first step toward uncovering the character of the causal relationship between two variables. So, for example, in the literature on Southern politics, the causal relationship between black mobilization and Republican growth (at the local, state, or regional level) is unclear and under-analyzed. Granger tests would help clarify this relationship (as we demonstrate in an extended example below).

The development of multivariate vector autoregressive (VAR) Granger models was a significant response to problems associated with bivariate Granger models. The rudimentary bivariate Granger procedure is easily extended to a VAR framework. Given a set of variables, one can test for the existence of causal relationships between each pair of variables, and variables that are not Granger caused by any other variable in the system are described as weakly exogenous (see Greene 2000). Suppose we were interested in the causal interrelationship between economic growth, income inequality, and democratization. If we could show that neither economic growth nor income inequality Granger caused democratization, then democratization would be weakly exogenous. Note that this is a relatively stringent exogeneity condition; it precludes, for example, the possibility that a nonrecursive relationship exists between two variables. This is not the only problem with Granger VAR models. VAR time series are often short, and complicated multivariate VAR models quickly overwhelm the available df in single time series.³ To address this problem, econometricians have begun to modify Granger tests to incorporate panel and TSCS dynamics (see e.g., Holtz-Eakin, Newey, and Rosen 1988; Arellano and Bond 1991; Hurlin and Venet 2001; Hurlin 2005).⁴ Within TSCS frameworks, Granger tests generate meaningful results with significantly shorter time spans, incorporate significantly more observations, and produce more efficient results than Granger tests in conventional contexts (Hurlin and Venet 2001).

Employing conventional Granger tests with TSCS data, however, raises two important inferential issues, both dealing with the potential heterogeneity of the individual cross-sections. One form of cross-sectional variation is due to distinctive intercepts, and this type of variation may be addressed with a fixed effects model.⁵ The other, arguably more problematic type of heterogeneity—causal variation across units—requires a more complex analytical response.⁶ Until recently, this type of heterogeneity was largely ignored (with unknown results) in TSCS VAR models. According to Erdil and Yetkiner (2005), there are two distinctive literatures dealing with TSCS or panel VAR models. One, the literature based on early work by Hsiao (1986) and Holtz-Eakin, Newey, and Rosen (1988) largely ignores this type of heterogeneity. So, for example, tests for weak exogeneity based on this literature did not (and do not) effectively account for the possibility of causal variation among the cross-sections; so it was not possible to know for which set of cross-sections the exogeneity result actually applied.⁷

Another strain of this literature, which was based on recent work by Hurlin and Venet (2001), explicitly addresses this type of heterogeneity. Hurlin and Venet (2001) outline a procedure for evaluating the character of the causal processes (homogenous versus heterogenous) within a TSCS framework.⁸ Their methodology is an extension of the basic Granger model. In this conventional context, series x may be said to cause series y if and only if the expectation of y given the history of x, E(y|x_t-k), is different from the unconditional expectation of y, E(y).⁹ That is, if

(1)

then x has no effect on—and thus does not cause—y. However, if

(2)

we would conclude that x does cause y. The expected value of y is different given x, so this implies that x causes y.

Extension of Hurlin and Venet (2001) on this basic model to the TSCS context is straightforward (but not trivial). Their analytical results are based on a model of the following type,

(3)

for each of the cross-sections i and for all t in [1,T]:¹⁰ Here we have a TSCS model where the regressors are (1) lagged values of the dependent variable (y_i,t–k) subset by cross-sections i and (2) lagged values of the independent variable (x_i,t–k) also subset by cross-section. The error term is represented by v_i,t. ^(k) are the autoregressive coefficients, and β are the regression coefficients. The fixed effects are represented by _i. The number of time periods is indicated by p. To retain sufficient degrees of freedom for model estimation, Hurlin and Venet assume the following constraints:

1. ^(k) are constant and
   
2. β are assumed constant for all k [1, p].
   

The first assumption precludes variation in the autoregressive coefficient from cross-section to cross-section. The second assumption precludes variation in the regression coefficients from time period to time period.¹¹ In addition, whereas the autoregressive slope coefficients are identical for all cross-sections, the regression coefficients are allowed to vary across individual cross-sections. Within the TSCS framework, there are three possible causal scenarios:

1. An identical causal relationship exists between x and y in all cross-sections.¹²
   
2. No causal relationship exists between x and y in any cross-section.
   
3. There is a causal relationship between x and y in some subset of n cross-sections, but the nature of the relationship is not constant across cross-sections.¹³ Hurlin and Venet subdivide this scenario into two categories: (a) causal relationships of different types across cross-sections and (b) at least one cross-section shows no evidence of any causal relationship.
   

The formalization Hurlin and Venet's formalization of the first scenario in the TSCS Granger framework—that there is no evidence that x causes y in any of the cross-sections—implies that

(4)

for every cross-sectional unit in the panel. E(y_i,t|y_i,t–1,y_i,t–2,...y_i,0,x_i,1,x_i,t–1,x_i,t–2,...x_i,0,_i) is the best linear predictor of y_i,t given the past values of y for each cross-section and the past values of x for each cross-section.¹⁴ According to equation (4), the expected value of y given past values of y is equal to the expected value of y given both past values of y and past values of x for each cross-section i. This is a straightforward extension of the null hypothesis for the standard Granger model; knowing the previous values of the independent variable x provides no new information about the current value of the dependent variable y that is not already contained in the previous values of y. This equation merely states that this is true for each and every cross-section in the TSCS.

Alternatively,

(5)

implies the existence of a causal relationship for every cross-sectional unit in the data. In contrast to equation (4), this inequality indicates that the expected value of y depends upon both past values of y and past values of x for each cross-section i. Thus, we can reject the TSCS version of the Granger null hypothesis; knowing the previous values of the independent variable x provides new information about the current value of the dependent variable y that is not already contained in the previous values of y. This equation also indicates that the inequality holds for all cross-sections in the TSCS. Note that each individual linear predictor is assumed constant, which indicates that the nature of the causal relationship is identical across all cross-sections.¹⁵

The third scenario allows for the presence of at least one causal relationship (up to a maximum of N not necessarily identical causal relationships) but does not require the existence of N causal relationships—heterogeneous causality. So, heterogeneous causality requires that equation (5) holds for at least one cross-section i in the data. The final case refers to that situation in which equation (4) holds for at least one cross-section i in the data. This would indicate that for at least one cross-section, there is no causal relationship. This exhausts the theoretically possible outcomes associated with TSCS Granger tests. We now move to a more detailed description of the implementation of the Hurlin and Venet procedure for empirically distinguishing between these various cases.¹⁶

	3. A Method for Causality Testing in TSCS Settings

Top 1. Introduction 2. Assessing Causality--The... 3. A Method for... 4. Substantive Application: The... 5. Discussion of Results 6. Future Directions 7. Conclusion Notes References

 
The methodology of Hurlin and Venet for evaluating causal heterogeneity in TSCS data comprise a set of nested F-tests. F-tests are standard tools for assessing the significance of parameter restrictions in linear models; more specifically, they provide a means for estimating the likelihood that one or more parameter restrictions significantly affect a model's goodness of fit.¹⁷ The procedure described below is a series of particular F-tests. These tests are easily implemented using the constrained regression technique in Stata® or in similar programs such as E-Views®. Before outlining the specific hypotheses and requisite tests, we outline the basic process below:

1. conduct TSCS nonstationarity tests for each variable to be examined,¹⁸
   
2. create a dummy variable to represent each cross-section in the sample,
   
3. create a set of slope parameters by multiplying each unit-specific variable by lags of the independent variable,
   
4. specify the necessary equations with the proper constraints, excluding the constant term,
   
5. for each model, record the sum of squared residuals or save it as a new variable, and
   
6. calculate the required test statistics and determine significance.
   

A flowchart outlining the nested testing procedure described in the previous section is presented in Fig. 1.¹⁹ The first test to be conducted is designed to test the proposition that x does not cause y for any cross-section in our sample. Rejection of this hypothesis implies that for at least one (and possibly more) cross-section, x Granger causes y.²⁰ Formally, the first hypothesis with cross-sections denoted as i is as follows:

View larger version (14K): [in this window] [in a new window] [Download PowerPoint slide]   Fig. 1 Granger causality testing for TSCS data.

 

H₁:
For all i, x does not cause y.

We assess this first hypothesis by constructing a test statistic, hereafter referred to as F₁, comparing the sum of squared residuals from a set of restricted models (RSS₂) to the sum of squared residuals produced by a set of baseline (unrestricted) models (RSS₁). As in the traditional Granger causality test, the unrestricted model includes lags of y_i,t–k, lagged values of the independent variable (x_i,t–k), and the fixed effects themselves (_i) to predict current values of y_i,t. Lagged values of the dependent variable are constrained to be equal (_i,t–1 = _i,t–k) for all models presented. In the unrestricted model, subsequent lags within-TSCS slope coefficients are also set to be equal (β_i,t–1 = β_i,t–k). In the restricted model slope coefficients and lags are constrained to 0 (β_i,t–1 = 0), leaving only the unit specific effects and the various lags of the dependent variable 0 to predict current values of y.²¹ The test statistic to determine the presence of causality (F₁) is calculated as:

where N is the number of cross-sections, p is the number of lags, and T is the number of time periods. Interpretation of the statistic relies on the F-distribution with Np, NT – N(1 + p) –p df. An insignificant statistic for this test indicates that x does not cause y in any cross-section, and thus, the testing process ends here. A significant test statistic indicates that for at least one (and possibly all) of the cross-sections, there exists a causal relationship and we proceed to a test for homogenous causality.

Having determined that x does cause y for i 1, we now proceed to determine whether a common (or homogenous) causal process exits for all i. The collective nature of the causal process is determined by testing the following hypothesis:

H₂:
For all i, x causes y.

In order to test H₂, we calculate another test statistic, hereafter referred to as F₂. Acceptance of H₂ (denoted by an insignificant test statistic) indicates that a common causal process is manifest for all cross-sections in our sample. At this juncture, further testing is unnecessary as x is said to Granger cause y for all TSCS cross-sections. Rejection of H₂, denoted by a significant test statistic, would indicate that for at least one or more TSCS cross-sections, x does not Granger cause y. The F₂ test statistic is calculated using the sum of squared residuals from the unrestricted model described above (RSS₁) along with the sum of squared residuals (RSS₃) from a restricted model in which the slope terms are constrained to be equal for each cross-section in the sample (β_t–1 = β_t–k). Calculation of the F₂ test statistics is as follows:

Again, the test statistic is interpreted using an F-distribution with Np, NT – N(1 + p) –p df.

If H₂ is rejected, a third hypothesis can be utilized in order to determine for which cross-section i, x causes y or:

H_3a:
For i, x does not cause y.

For each cross-section i, H_3a is conducted using the unrestricted sum of squared residuals estimated previously (RSS₁) in addition to the sum of squared residuals (RSS_2,i) from a model in which the slope coefficient for the cross-section in question is constrained to 0 or excluded from the model equation (β_i,t–k= 0). The statistic to test H_3a for cross-section i is calculated as follows:

If the F_3a test statistic is significant using a F-distribution with Np, NT – N(1 + p) – p df, then we can reject H_3a, indicating that x does cause y for cross-section i. If the F_3a test statistic is insignificant, then x does not cause y for cross-section i. Testing of H_3a is conducted for each TSCS cross-section across the requisite number of time periods (t – k) in order to determine which cross-sections contributed to the previous finding of causality (denoted by rejection of H₁).

When theory suggests grouping cross-sections in a specific manner, a second test statistic can be calculated to examine the causal nature for some subset of panel members j, or:

H_3b:
For j, x does not cause y.

In this case, the slope coefficients for the subset of cross-section members j in question are constrained to 0 (β_j,t–k = 0). The sum of squared residuals from this restricted equation (RSS_2,j) is again compared to those from the unrestricted model (RSS₁) to produce the F_3b test statistic:

where n_nc is the number of TSCS cross-sections for which β is constrained to 0 and n_c is the number of TSCS cross-sections for which β is not constrained to 0. At this juncture, a significant F_3b test statistic allows for the rejection of H_3b for subgroup j, inferring that x Granger causes y in this subset of cross-sections. We now apply this TSCS Granger procedure to test one recent theory of political development in the post-World War II South.

	4. Substantive Application: The Transformation of the Southern Party System

Top 1. Introduction 2. Assessing Causality--The... 3. A Method for... 4. Substantive Application: The... 5. Discussion of Results 6. Future Directions 7. Conclusion Notes References

 
Though dramatic, the rise of Southern Republicanism and the electoral mobilization of Southern blacks were not unexpected. Key's (1949) work alludes to this transformation, and by the end of the 1970s, both dynamics were the foci for a considerable body of scholarship (for a description of this literature, see Black and Black 2002). Undoubtedly, national political dynamics played an important role in the formative stages of the development of Southern Republicanism (see Black and Black 1987, 1992, 2002; Carmines and Stimson 1989), but national level forces cannot fully explain the subregional variation in GOP growth or the electoral mobilization of Southern blacks.²²

Explanations of Republican growth have focused primarily on demographic and economic factors such as immigration, the transformation and growth of the Southern economy, the growth in religious conservatism, and racial context. Recently, Hood, Kidd, and Morris (2004) have added a new wrinkle to the literature by suggesting that black mobilization drove Republican growth in the region. We argue that as blacks mobilized and became Democrats, it became increasingly difficult for white conservatives to maintain control of the party. In increasing numbers, white conservatives left for the relatively more attractive Republican Party (see Heard 1952). As conservative whites left the Democratic Party, the opportunities for Southern blacks in the Democratic Party increased. With the Democratic Party an increasingly attractive option, southern blacks mobilized in greater numbers. In short, the usefulness of the Republican Party increased over time for white conservative voters as the attractiveness of the Democratic Party decreased. We also argue that this mobilization dynamic is most likely to play out in those Southern states with the largest black populations—especially those with black populations large enough (if mobilized) to control the Democratic Party in an increasingly Republican environment.²³

To investigate the potentially endogenous relationship between GOP growth and black mobilization in the South, we turn to a TSCS Granger analysis to examine the extent to which: (1) GOP growth Granger causes black mobilization and (2) black mobilization Granger causes GOP growth for the Southern states from 1960 to 2004. We provide a fuller description of our data in the next section.

4.1 Variable Operationalization
For this study, the former states of the Confederacy serve as our unit of analysis, producing a total of 11 cross-sections. For each Southern state, we have collected biennial data over a 44-year time period, from 1960 through 2004, for two variables.²⁴ The first of these variables, Black Mobilization, taps into the potential influence that the political mobilization of blacks may have produced in regard to politics in the region. Black Mobilization is calculated at the state level as the number of black registered voters divided by the total number of registered voters, or:

Operationalized as it is, our measure of black electoral strength places blacks within the context of the existing electorate—a much more precise method for estimating the potential influence of blacks as an electoral presence than alternative indicators (i.e., the percentage of blacks registered to vote).

The second indicator of interest, GOP Strength, is measured at the state level utilizing an index developed by David (1972). General election vote percentages for Republican candidates in gubernatorial, senate, and congressional elections were utilized to create a composite state-level index of GOP strength.²⁶ Following the construction of each GOP state index, a 10-year (5-time point) moving average was applied to smooth any sharp variations present in each series.²⁷ The David Index of Party Strength was the method of choice for Lamis (1988) in his detailed study of party change in the South. Specifically, GOP Strength is calculated as:

These two series are plotted by state from 1960 through 2004 in Fig. 2, with GOP Strength represented by black squares and Black Mobilization represented by white triangles. As evidenced by this set of 11 plots there is a great deal of variation, both temporal and cross-sectional, for each of these series. The minimum and maximum values for GOP Strength are 0.01 (Mississippi-1960) and 0.57 (Tennessee-2004), respectively. For Black Mobilization, the minimum value is 0.04 found in 1960 Mississippi, whereas the maximum value of 0.38 is also associated with Mississippi in 1994.

View larger version (18K): [in this window] [in a new window] [Download PowerPoint slide]   Fig. 2 GOP Strength and Black Mobilization by state, 1960–2004.

 
4.2 Testing for Nonstationarity
Before proceeding with the TSCS Granger tests outlined above, we need to establish that both pooled time series are stationary (do not contain a unit-root). We utilize two different tests designed to detect the presence of unit-roots specifically in TSCS data. Table 1 presents test statistics from the Levin, Lin, and Chu and the Im, Pesaran, and Shin procedures, both of which indicate that nonstationarity is not an issue for either series.²⁹

View this table: [in this window] [in a new window]   Table 1 TSCS unit-root tests

 
4.3 Causality Testing
The first step in untangling the causal process between the two variables of interest is to test H₁. In our case, we want to know if:

for the 11 states in our sample. In addition, if a causal relationship is detected, we need to determine its consistency across the states in our sample. We begin by testing H₁, which is:

H₁:
For all states, GOP Strength (Black Mobilization) does not cause Black Mobilization (GOP Strength).

In order to test H₁, we calculated the F₁ test statistics using the sum of squared residuals from the unrestricted models:

(9)

(10)

where the slope coefficients for subsequent lags within states are constrained to be equal (i.e., ) and a second model identical to the one specified in equation (9) (equation 10), where the slope coefficients and subsequent lags for all the states in our sample are constrained to 0 (i.e., β_i,t–1 ... β_i,t–k = 0). Again, a significant F₁ test statistic indicates that for at least one (and possibly all) of the states in our analysis, Black Mobilization Granger causes GOP Strength or GOP Strength causes Black Mobilization.

The F₁ test statistics are presented in Table 2. The first test of H₁ analyzes whether GOP Strength, for the members of our TSCS collectively, Granger causes Black Mobilization. The F₁ test statistic is statistically significant at one lag t–1, allowing us to reject H₁. So for at least one state (and possibly all), there is statistical evidence that GOP Strength Granger causes Black Mobilization. The F₁ test statistics, however, are insignificant at two and three lags, an indication that there is little evidence to support the claim that this relationship exists beyond a single lag period.

View this table: [in this window] [in a new window]   Table 2 F1 test statistics

 
The second half of Table 2 details the F₁ test statistics used to examine the hypothesis that Black Mobilization Granger causes GOP Strength. At one lag, the F₁ test statistic is significant, indicating again that this causal process is at work in one or more of the states in our sample. Again, there appears to be no evidence that this relationship exists in subsequent time periods (i.e., t–2, t–3). In summary, we can reject H₁ for the relationships under study and proceed next to determine the nature of the causal process across the 11 states of the South.

The following hypothesis is designed to examine the homogeneity of the causal relationship uncovered in the preceding step:

H₂:
For all states, GOP Strength (Black Mobilization) causes Black Mobilization (GOP Strength).

In order to test H₂, we calculate another set of test statistics, hereafter referred to as F₂, using the sum of squared residuals from the unrestricted model specified above and those from a model where the slope coefficients are constrained to be equal for all states and lag periods (i.e., β_1,t–1 = β_2,t–2 = β_n,t–k). Failure to reject H₂ (insignificant test statistic) indicates that the causal process is homogenous for all 11 states in our sample. At this juncture, if H₂ is confirmed, further testing is unnecessary as Black Mobilization is said to Granger cause GOP Strength (or GOP Strength is said to Granger cause Black Mobilization) for all the states in our sample. On the other hand, rejecting H₂ (significant test statistic) would indicate that for at least one or more states n 1, Black Mobilization does not Granger cause GOP Strength (or GOP Strength does not Granger cause Black Mobilization).

The results of our tests for H₂ are located in Table 3. Again, the bidirectional relationship between Black Mobilization and GOP Strength is examined for up to three lag periods. In the case of GOP Strength Granger causing Black Mobilization, H₂ is rejected at one lag. Therefore, we must conclude that the causal process in this case is heterogeneous or does not exist across all the states in our sample. Reversing the equation, however, we find that H₂ is accepted in the case of Black Mobilization Granger causing GOP Strength. So, for this relationship, the causal process appears to be homogenous across all 11 states, making further examination unnecessary.

View this table: [in this window] [in a new window]   Table 3 F2 test statistics

 
In order to determine which states in our sample contribute to the causal finding leading from GOP Strength to Black Mobilization, we must turn to a third hypothesis or H_3a which can be formally stated as:

H_3a:
For state i, GOP Strength does not cause Black Mobilization.

Rejection of H_3a indicates the presence of a causal relationship for the specific state under consideration. The F_3a test statistics are calculated using the sum of squared residuals from the unrestricted model in equation (1) and a second model in which the slope coefficients and subsequent lags for the state under testing are constrained to 0 (i.e., F_3a: β_i,t–1 ... β_i,t–k = 0). We present state-by-state results in the upper portion of Table 4 for one lag period t–1.³⁰ The test results indicate that for Alabama, Georgia, Louisiana, Mississippi, North Carolina, and South Carolina the relationship GOP Strength Granger causes Black Mobilization appears to hold. For the remaining five states in our study there is insufficient statistical evidence to reject H_3a.

View this table: [in this window] [in a new window]   Table 4 F3 test statistics: GOP Strength Granger causes Black Mobilization(t-1)

 
What one quickly notices about these two sets of states is that they divide nicely into two well-identified geographic subregions: deep South and rim South. The sole exception is North Carolina, typically considered a rim South state. The bottom portion of Table 4 tests H_3b by grouping states into their respective subregion. By convention, we classify North Carolina as a rim state along with Arkansas, Florida, Tennessee, Texas, and Virginia with the remaining states comprising the deep South. In this example, H_3b is specified as:

H_3b:
For subregion j, GOP Strength does not cause Black Mobilization.

In order to determine whether GOP Strength ganger causes Black Mobilization for each subregion, we derive a set of test statistics (F_3b) using the sum of squared residuals from equation (1) and from a second model in which the slope coefficients for the states comprising the subregion are constrained to 0 (i.e., for the deep South: F_3b: β_j,t–1 = 0).³¹ As indicated in Table 4, H_3b is rejected collectively for the deep South states and accepted for the rim states. Thus, testing by subregion, we may conclude that at t–1, GOP Strength Granger causes Black Mobilization in the deep South, but not in the rim South.

	5. Discussion of Results

Top 1. Introduction 2. Assessing Causality--The... 3. A Method for... 4. Substantive Application: The... 5. Discussion of Results 6. Future Directions 7. Conclusion Notes References

 
In summary, we find that Black Mobilization Granger causes GOP Strength and that GOP Strength Granger causes Black Mobilization. The former causal process can be characterized as homogenous for the 11 states in our sample. The later process, conversely, is heterogeneous, existing in the deep South (and North Carolina), but not in the rim states. In the case of the rim South, the causal process is one-sided and can be characterized as:

In the case of the deep South states, the process is reciprocal:

and

Not only is there a geographical differentiation associated with these subregions, but a substantive one as well. Although the size of the black population is much higher in the South compared to other regions of the United States, within the South the highest concentrations of black citizens are located in the deep South states. This fact becomes clear looking at Fig. 3, which arrays the 11 states in the region according to the relative size of their average black populations from 1960 through 2000. The average black population in the region varies from a high of 37% in Mississippi to a low of 12% in Texas.

View larger version (32K): [in this window] [in a new window] [Download PowerPoint slide]   Fig. 3 Mean Black population by state, 1960–2000.

 
The horizontal line in Fig. 3 separates states based on the significance of their F_3a test statistics in Table 4. These test statistics were significant for those states above the horizontal line and insignificant for the reminder located below this division. This line also suggests a potential threshold for the noted counter-mobilization effect to manifest itself. In states with an average black population over 20%, expansion of the Republican Party produced increased levels of mobilization within the black electorate. Below this level, it is possible that a necessary critical mass does not exist to trigger the noted counter-mobilization reaction found in the deep South states.

As Key noted decades ago:

The range of the Negro population—from 49.2 percent in Mississippi to 14.4 percent in Texas—suggests that even "the South" is by no means homogenous and that if the Negro influences the politics of the South, there ought to be wide variations in political practices from state to state (1949:10–11).

This variation has traditionally been thought of in quantitative terms. The example utilized suggests that a qualitative distinctiveness also exists. These results, taken in tandem, suggest the transformation of Southern politics during the last half of the twentieth century can be viewed from a theoretical perspective that attributes the same logic of action to both blacks and conservative whites. This is a simple and straightforward feedback loop (in over half of the Southern states), but it is also a novel characterization of the twin pillars of the transformation of Southern politics: Republican growth and black mobilization.

With few exceptions, research on Republican growth in the South has focused on other types of causal explanations including economic dynamics, religious or cultural orientations, migration patterns, or the geographic concentration of blacks (i.e., the black-belt hypothesis), while largely overlooking more overtly political explanations. Similarly, existing work on black mobilization in the South rarely focuses explicitly on the party dynamics highlighted in this example.

In an important respect, our results are consistent with the black empowerment literature (Browning, Marshall, and Tabb 1984; Bobo and Gilliam 1990; Harris, Sinclair-Chapman, McKenzie 2005). However, instead of focusing on the outcome produced by black mobilization—the election of black officials (in the South)—and then viewing that outcome as an inducement to further black mobilization, one might explain our results with a more fundamental political force: the change in the benefits for blacks of Democratic Party membership and activism (and, implicitly, electoral mobilization). Clearly, Republican growth (and the exodus of white conservatives from the Democratic Party) opened up significant opportunities for blacks.³² Our research suggests that Southern blacks saw these opportunities—where they were greatest—and took advantage of them.

	6. Future Directions

Top 1. Introduction 2. Assessing Causality--The... 3. A Method for... 4. Substantive Application: The... 5. Discussion of Results 6. Future Directions 7. Conclusion Notes References

 
There is considerable evidence of a nonrecursive relationship between the two factors of interest; however, this two-way causal flow is not homogenous across all the states in our sample. Again, for the rim South states there is no endogeneity issue, as Black Mobilization was found to Granger cause GOP Strength, with no evidence for the converse. For those states located in the deep South, the TSCS Granger tests did point to evidence of a nonrecursive relationship between these two variables of interest.

The logical next step would involve specifying a set of models within a multivariate framework in order to determine if the causal relationships uncovered stand up to the addition of control variables. In formulating multivariate explanatory models, the best course of action would call for separate analyses, one for deep South states and another for the rim South states. For the former, instrumental variables will need to be employed to alleviate known problems associated with the use of endogenous regressors, whereas for the latter a more traditional TSCS regression framework can be utilized (for a discussion of panel models using instrumental variables, see Baltagi 2005).

	7. Conclusion

Top 1. Introduction 2. Assessing Causality--The... 3. A Method for... 4. Substantive Application: The... 5. Discussion of Results 6. Future Directions 7. Conclusion Notes References

 
This manuscript examines a new methodology for evaluating the causal homogeneity assumption in TSCS data via the use specially designed Granger causality tests. The TSCS Granger tests presented allow one to control for, and detect, the possibility that the causal process, if present, is not homogenous for members of the cross-section under study (i.e., heterogenous causality). As the utilization of TSCS data for the exploration of political phenomena has grown exponentially over the last decade, this diagnostic device should enjoy widespread application.

Policy studies exploring the relationship between concealed weapons and crime over time and across some geographic unit are one example where these specialized Granger tests might be employed. The study of Kovandzic and Marvell (2003) attempts to determine the effect that concealed weapon permits produce on the rate of various types of violent crime in Florida. The pair find little evidence that the rate of concealed weapon permits within a county act to depress violent crime rates. It is possible, however, that a nonrecursive relationship may exist in which increases in the violent crime rate may cause an increase in citizens seeking to obtain concealed weapon permits. Likewise, the finding that permits are not related to a decrease (or increase) in violent crime rates may not be valid across all the Florida counties sampled. It is possible that a relationship between these two factors may be heterogeneous, present for some subset of counties but not for others.

Shaw's (1999) examination of campaign-specific effects on presidential voting patterns provides another example where this methodology might be employed. The Republican share of the two-party vote, by state, is estimated on a weekly basis using candidate appearances and campaign advertising as explanatory factors. The included campaign effects were found to be positively related to a candidate's vote share within a state. The possibility also exists, however, that campaigns may adjust to a changing political environment. In this case, vote share (or estimated vote share) may cause the frequency of advertising or appearances to shift. Likewise, one might envision a situation where campaigns affect vote share, but not in a uniform manner across all states. States with higher numbers of independents may be more apt to be influenced by campaign effects, compared to states with lopsided partisan divisions.

Although time-series, cross-sectional research has become prevalent in political science scholarship, diagnostic tests and other extensions to deal with known TSCS issues have sometimes been slow to develop. Recent research indicates that one known problem—the questionable assumption of causal heterogeneity—is far more prevalent than previously realized. We agree with the conclusion of Wilson and Butler that there is a clear "need for extensive sensitivity testing as part of the research process," (2007: 119) especially where TSCS data are involved. The modified Granger tests presented in this manuscript offer one possible tool to help researchers evaluate the extent of causal heterogeneity within a TSCS data set. Such information, in turn, can then be used to specify more accurate multivariate time-series cross-sectional models.

Authors' note: We would like to thank Christophe Hurlin and Baptiste Venet for their path-breaking work in this area and especially for the helpful comments provided by Professor Hurlin. In addition, we would also like to thank Geoff Layman, Mike Hanmer, and the participants of the American Politics Workshop at the University of Maryland, College Park for their helpful advice. A previous version of the manuscript was presented at the 2006 Citadel Symposium on Southern Politics.

	Notes

Top 1. Introduction 2. Assessing Causality--The... 3. A Method for... 4. Substantive Application: The... 5. Discussion of Results 6. Future Directions 7. Conclusion Notes References

 
¹ The use of Granger analysis in political science is extensive. Recent uses of Granger techniques range across the discipline and focus on substantive questions as diverse as the causal relationships between democracy and economic growth (Heo and Tan 2001; Baum and Lake 2003), partisanship and ideology in the American electorate (Box-Steffensmeier and De Boef 2001), domestic politics and foreign policy (Moore and Lanoue 2003), school performance and the size of educational bureaucracies (Meier, Polinard, and Wrinkle 2000), and terrorist incidents and the threat of terrorism (Enders and Sandler 2000).

² We define k as the number of lagged values of variables x and y.

³ These problems are discussed in some detail in Kmenta (1997) and Greene (2000).

⁴ Some substantive examples of this methodology include Podrecca and Carmeci (2001), Weinhold and Nair (2001), Davis and Hu (2004), Hurlin and Venet (2004), and Erdil and Yetkiner (2005). Though political scientists commonly distinguish between panel data (more cross-sections than time periods) and TSCS data (more time points than cross-sections), economists often do not. The procedure presented below is most appropriate for TSCS data, so we use the TSCS terminology. However, it is also appropriate for panel data with a manageable number of cross-sections (i < 50 in most cases) and sufficiently long time series for modeling temporal dynamics (t > 10).

⁵ For example, suppose black mobilization has the same causal impact on Republican growth throughout the South. That would not preclude variation in Republican growth across the Southern states due to other idiosyncratic factors outside the model. Fixed effects models are one way to address this type of cross-sectional variation.

⁶ This causal heterogeneity should not be confused with the "unobserved heterogeneity" common in censored TSCS data or in TSCS data for which sample selection bias is a potential problem. In those situations, one or more unobservable variables (unobservable for practical or theoretical reasons) is correlated with (or is potentially correlated with) an observed independent variable. This correlation may lead to spurious results, attributing a causal effect to an observed variable because the unobserved variable cannot be incorporated into the model. In this case, there is no expectation of causal heterogeneity; if all appropriate variables were observable, the same causal model would be appropriate for all observations—that is, causal homogeneity. The various solutions to "unobserved heterogeneity" (see, e.g., Heckman 1981; Vella and Verbeek 1998, 1999; Wooldridge 2000, 2002, 2005) are based upon this substantive assumption. Our concern is with the appropriateness of the assumption of causal homogeneity in the first place.

⁷ This may be partly attributable to the preponderance of large i, small t data sets in applied econometrics. Panels with hundreds of cross-sections and 10 or fewer time points are very common. Given that data structure, tests for causal heterogeneity—even the tests developed by Hurlin and Venet—would be difficult to implement. However, in political science—and increasingly in economics—TSCS data sets with significantly fewer cross-sections and significantly more time points are quite common. It is these types of data sets for which the tests for causal heterogeneity are particularly appropriate.

⁸ Technically, this is also a TSCS VAR model, but it is limited—as currently developed by Hurlin and Venet—to the evaluation of bivariate relationships.

⁹ The independent variable is denoted by x and the dependent variable by y. The time period is indicated by t, and the number of lags is indicated by k.

¹⁰ Note that the residuals satisfy the conventional assumptions. See Hurlin and Venet (2001) for further details.

¹¹ Note that regressions coefficients are allowed to vary across lag lengths.

¹² Situations in which a causal relationship is present in each cross-section but the character of the causal relationship is variable are included in Scenario 3.

¹³ Heterogeneity arising from level differences between cross-sections is addressed by including unit-specific (fixed) effect parameters. This is an imperfect strategy. If we were willing to assume the absence of true causal heterogeneity—attributing all cross-sectional heterogeneity to sampling error—then an array of alternatives to fixed effects would be available to us (for a thorough discussion of these options, see Wooldridge 2002). The development of alternatives to fixed effects would be a welcome—but still as yet unrealized—extension of the procedure presented here.

¹⁴ Hurlin and Venet (2001) explicitly define the best linear predictor for the instantaneous case (includes the current value of x as a regressor). Our definition of the best linear predictor for the more conventional Granger case (does not include the current value of x as a regressor) is a straightforward extension.

¹⁵ See original constraints on autoregressive coefficients and regression coefficients.

¹⁶ Effectively distinguishing between substantive differences between cross-sections and differences resulting from sampling error depends upon a sufficiently long-time series. For this reason, these procedures are most appropriate for TSCS data and are not appropriate for panels with particularly short time series (t < 10).

¹⁷ Note that the F-test tends to be the most conservative of a set of analogous tests (i.e., Wald, LM, and LR tests) and is the least sensitive (of these tests) to small sample deviations from normally distributed disturbances (see Greene 2000).

¹⁸ These models operate under the assumption that all series to be tested do not contain a unit root. The crucial requirement is covariance stationarity, and none of the results require more than weakly stationary series. Popular statistical software packages (such as Stata) include standard tests to assess stationarity in TSCS data. Two examples would be the Levin, Lin, Chu Test or the Im, Pesaran, and Shin Test. See Baltagi (2005) for a discussion of unit root tests for TSCS data. A potentially useful extension of this methodology to a context including partially integrated series remains undeveloped. Although some procedures do exist for evaluating the level of integration of partially integrated series (i.e., the Robinson log periodogram estimator [roblpr in Stata]), it is not clear how to most effectively account for time series that manifest different levels of partial integration—a likely possibility in TSCS data sets in which there is a serious question about the comparability of the cross-sections (for further discussion of diagnostic tests for fractional integration, see Baum and Wiggins 2001).

¹⁹ Nomenclature and equations come directly from Hurlin and Venet (2001).

²⁰ It is important to note that rejection of this hypothesis does not necessarily imply the presence of a homogenous causal process for the entire TSCS data set (or that x causes y for all cross-sections).

²¹ Slope coefficients for x can be either constrained to zero or excluded from the model altogether.

²² Hood, Kidd, and Morris (2004); Nadeau and Stanley (1993); Rhodes (2000); and Shafer and Johnston (2001) provide useful overviews of this literature.

²³ These would be the deep South states or Alabama, Georgia, Louisiana, Mississippi, and South Carolina.

²⁴ Our time series consist of 2-year election cycles (i.e., 1960, 1962, ... 2004).

²⁵ Interpolation was used to fill in the gaps between missing years for both the number of blacks who were registered to vote and for the total number of registered voters in each Southern state. [Data sources: VEP News (Various Years); Statistical Abstract (Various Years); Current Population Reports: P-20 Series on Voting and Registration (Various Years)].

²⁶ Estimates from 1960 through 1970 are obtained from David's work, whereas estimates for the remaining years are calculated by the authors. [Data sources: David (1972); Guide to U.S. Elections (1994); America Votes (Various Years)].

²⁷ Comparisons between our measure of Republican Party strength and actual party registration data from Louisiana and Florida from 1950 to 2000 (the only two Southern states that did track party registration during the time of our study) indicate a high level of congruity (r = 0.94 for LA and 0.94 for FL) [data available from the authors upon request].

²⁸ Special transformations had to be made for Louisiana for each election following the 1978 institution of an open primary system. We used the following method to calculate our index of GOP party strength for 1978 through 2004:

1. 1. If there was only one election (open primary): GOP = percent of total republican vote (including votes won by other Republican candidates in the primary).

2. 2. If there was both a primary and a general election, and

   a. The general election contained both a Republican and a Democrat: GOP = percent of total vote won by Republican Candidate.

   b. The general election contained two Democratic candidates: GOP = 0%,

   c. The general election contained two Republican candidates: GOP = 100%.

²⁹ We also conducted unit-root tests for the raw (unsmoothed) GOP Strength series. The Levin, Lin, and Chu (test statistic: –4.60; p < .001) and the Im, Pesaran, and Shin (test statistic: –2.77; p < .001) tests both indicate this series is also stationary.

³⁰ There is little evidence that this causal relationship exists beyond the first lag period (see Table 2). We therefore restrict tests of H_3a to t–1.

³¹ In this case, we only calculate the F_3b test statistic to one lag t–1.

³² For example, in 2004, blacks comprised 58% of the turnout in the South Carolina Democratic Primary and 47% in the Georgia Democratic Primary (Bullock and Gaddie 2005a; 2005b).

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